Abstract [en]

We consider the random interlacements process with intensity u on Z(d), d >= 5 (call it I-u), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on Z(d). For k >= 3 we want to determine the minimal number of trajectories from the point process that is needed to link together k points in I-u. Let n(k, d) := [d/2(k -1)] - (k - 2). We prove that almost surely given any k points x(1),..., x(k) is an element of I-u, there is a sequence of n(k, d) trajectories gamma(1),..., gamma(n(k,d)) from the underlying Poisson point process such that the union of their traces U-i =1(n(k,d)) Tr(gamma(i)) is a connected set containing x(1),..., x(k). Moreover we show that this result is sharp, i.e. that a.s. one can find x(1),..., x(k) is an element of I-u that cannot be linked together by n(k, d) - 1 trajectories.